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ONE–MEMORY IN MULTIPERSON BARGAINING

by

AYSU OKBAY

Submitted to the Graduate School of Arts and Social Sciences in partial fulfillment of the requirements for the degree Master of Arts

Sabancı University

July 2012

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ONE–MEMORY IN MULTIPERSON BARGAINING

APPROVED BY:

Mehmet Barlo ...

(Thesis Supervisor)

Ozge Kemahlıo˘ ¨ glu ...

Ozg¨ ¨ ur Kıbrıs ...

DATE OF APPROVAL: 20.07.2012

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Aysu Okbay 2012 c

All Rights Reserved

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Acknowledgements

I would like to start by thanking my thesis supervisor, Prof. Mehmet Barlo, for walking me through the whole thesis process without letting me get lost. I would not have been able to produce a work that I am this proud of without his invaluable guidance. I would also like to express my gratitude to him for making the process fun, and never losing, or letting me lose motivation.

My thesis jury members, Prof. ¨ Ozge Kemahlıo˘ glu and Prof. ¨ Ozg¨ ur Kıbrıs, de- serve my infinite thanks for both their valuable comments, and for raising my inter- est and knowledge in bargaining theory in the first place, along with Prof. Mehmet Barlo.

The love and support of my family should not go unnoticed. I really appreciate them for not badgering me with questions about the progress of my thesis.

I am very grateful to Canan Tahao¸ glu for her outright help and support whenever I needed.

Last but not least, I would like to thank Fırat for anything and everything.

Trying to raise my morale over the technologically impeded, impossibly difficult

Skype conversations could not have been easy. Moreover, I cannot thank enough the

magnificent Memik and Lucy Karakulak for always putting a smile on my face.

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ONE–MEMORY IN MULTIPERSON BARGAINING

Aysu OKBAY

Economics, MA Thesis, 2012 Thesis Supervisor: Mehmet BARLO

Keywords: Bargaining, recall, bounded rationality, complexity, stationarity.

Abstract

In Rubinstein’s (1982) 2–player discounted alternating offers bargaining game,

the subgame perfect equilibrium outcome is unique and equivalent to the Nash bar-

gaining solution. However, when there are more than 2 players, every feasible parti-

tion can be sustained in subgame perfect equilibrium with a sufficiently high discount

factor (Shaked 1986). We prove that when the restriction to one–memory strategies

is employed in the multiplayer version of the game, the subgame perfect equilibrium

is unique and equivalent to the multiplayer generalization of Rubinstein’s. This also

implies that the unique subgame perfect equilibrium outcome corresponds to the

Nash solution in the multiplayer cooperative game.

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C ¸ OK–K˙IS ¸ ˙IL˙I PAZARLIKTA B˙IR–HAFIZA

Aysu OKBAY

Ekonomi, Y¨ uksek Lisans Tezi, 2012 Tez Danı¸smanı: Mehmet BARLO

Anahtar Kelimeler: Pazarlık, hafıza, kısıtlı rasyonellik, karma¸sıklık, de˘ gi¸smezlik.

Ozet ¨

Rubinstein, 1982 tarihli makalesinde, 2–oyunculu iskonto edilmi¸s sıralı–teklif pazarlık oyununda, sadece tek bir alt-oyun-yetkin denge oldu˘ gunu, ve bu dengede elde edilen da˘ gılımın, iskonto de˘ geri 1’e yakla¸stı˘ gında, Nash pazarlık ¸c¨ oz¨ um¨ une denk oldu˘ gunu kanıtlamaktadır. Fakat, oyuncu sayısının 2’den fazla oldu˘ gu du- rumlarda, yeteri kadar y¨ uksek bir iskonto fakt¨ or¨ u ile m¨ umk¨ un olan her da˘ gılım, bir alt-oyun-yetkin denge sonucu olarak elde edilebilmektedir (Shaked 1986). Biz, bu

¸calı¸smada, bahsi ge¸cen oyunun ¸cok-ki¸sili versiyonunda izin verilen stratejilere bir–

hafıza kısıtlaması getirmekte, ve bu kısıt altında alt-oyun-yetkin dengenin tek, ve

Rubinstein’ın 2–ki¸sili oyunda elde etti˘ gi dengenin ¸cok-ki¸si versiyonları i¸cin yapılan

genellemesine e¸sit oldu˘ gunu ispatlamaktayız. Nitekim bu, aynı zamanda tek denge

da˘ gılımının ilgili ¸cok-ki¸sili i¸sbirlik¸ci oyundaki Nash ¸c¨ oz¨ um¨ une de y¨ uksek iskonto

de˘ gerlerinde denk oldu˘ gu anlamına gelmektedir.

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Contents

1 Introduction 1

2 The Model 6

3 The Bargaining Problem 9

3.1 Nash’s Barganing Solution . . . . 10

3.2 Rubinstein’s Result . . . . 11

3.3 The Multiplayer Case . . . . 14

3.4 Finite–State Automata and Bounded Rationality . . . . 16

3.5 The Loss of Game Theoretic Prediction Power . . . . 17

4 One–memory 19 4.1 Confusion . . . . 21

4.2 Shaked’s Strategy is not One–Memory . . . . 27

4.3 Stationary Subgame Perfect Equilibrium . . . . 28

4.4 One–Memory Subgame Perfect Equilibrium . . . . 30

5 Concluding Remarks 37

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List of Tables

3.1 Shaked’s SPE strategy profile σ

. . . . 15

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1 Introduction

A bargaining situation is described as the interaction between two or more players trying to reach an agreement among multiple alternatives, in which all of them would be at least as well off as the case of no agreement. Many interactions that economists are interested in, like negotiations between labor unions and employers, or actually almost any kind of trade between parties, can be modeled in this framework. This is why bargaining has continued to draw the attention of researchers over the years.

It was John Nash, who introduced game theorists to the subject with his pi- noeering paper “The Bargaining Problem” (Nash 1950), followed by “Two-player cooperative games” (Nash 1953). The significance of John Nash’s contribution is that he laid the groundwork for two different approaches to analyzing bargaining situations, which complement each other.

In the article titled “The Bargaining Problem”, Nash proposes an axiomatic approach to bargaining, which focuses on the “cooperative game”. To be more precise, he lists some axioms that seem natural for a solution to have, without specifying any details about the bargaining procedure. Then, taking as given only the set of all possible payoff profiles that can be obtained as the outcome of a bargaining situation, he finds out that the payoff profile satisfying all of these axioms is unique, which came to be known as the Nash bargaining solution.

In his latter paper, “Two-player cooperative games”, Nash describes a non-

cooperative bargaining game (known as Nash’s simultaneous-move demand game)

specifying all the details of the bargaining process, and analyzes the equilibrium

outcome, which turns out to be the same outcome as the cooperative solution. His

intuition is that non–cooperative games are more basic and analyzing cooperative

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games by solving for the equilibria of a corresponding non–cooperative game might provide a worthy insight:

We give two independent derivations of our solution of the two–person cooperative game. In the first, the cooperative game is reduced to a non-cooperative game. To do this, one makes the players’ steps in ne- gotiations in the cooperative game become moves in the non-cooperative model. Of course, one cannot represent all possible bargaining devices as moves in the non-cooperative game. The negotiation process must be formalized and restricted, but in such a way that each participant is still able to utilize all the essential strength of his position. The sec- ond approach is by the axiomatic method. One states as axioms several properties that would seem natural for the solution to have, and then one discovers that the axioms actually determine the solution uniquely.

The two approaches to the problem, via the negotiation model or via the axioms, are complementary. Each helps to justify and clarify the other.

(Nash (1953, p.128))

This attempt to bridge the gap between the two approaches was later called the Nash program. The unification of the two approaches provides justification for both, which they separately lack. As Sutton (1986) argues, the non–cooperative approach has the problem that the rules of a non–cooperative bargaining game might differ so extensively from case to case, that in order to obtain any useful insight into the theory, some principles that hold over a wide range of possible bargaining processes need to be prescribed. The axiomatic approach is helpful in this respect. However,

“prescribing” such principles, or axioms, normatively is also problematic in turn. In order to establish the reasonableness of these axioms, one must examine whether or not they hold in some plausible bargaining processes, and this is done by formulating some non–cooperative bargaining games and examining their equilibria.

As Serrano (2005) suggests, the basic motivation behind the Nash program is that the relevance of a concept is enhanced if one arrives at it from different approaches.

Thus, in the Nash program, the most significant research agenda is to find non–

cooperative bargaining games that yield the same, unique equilibrium outcome as

the cooperative Nash bargaining solution. In this regard, one of the most important

contributions was made by Rubinstein (1982), which examines a 2–player discounted

alternating offers bargaining game that uniquely sustains the Nash bargaining solu-

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tion in subgame perfection. What makes Rubinstein’s result significant is that unlike Nash’s simultaneous-move demand game which relies on uncertainty in obtaining the uniqueness of Nash equilibrium, Rubinstein’s uniqueness result comes from the use of credible threats, together with the assumption of impatience (Serrano 2005).

However, later Shaked (1986) proves that these two key features, the use of sub- game perfection and discounting, do not suffice to obtain a unique equilibrium in the 3–player version of the same game (restated in Osborne and Rubinstein (1990)).

In fact, he shows that every feasible outcome can be obtained in subgame perfect equilibrium when there are more than 2 players.

This striking result of Shaked displays that the game theoretic prediction power of the n ≥ 3 player generalization of Rubinstein’s game, collapses. Moreover, there is no link between the non–cooperative and cooperative solutions in the n ≥ 3 version of the game. The multiplicity of equilibria was tried to be eliminated using various extensions to the game. In Jun (1987), Chae and Yang (1988) and Fershtman and Seidmann (1993) players are restricted by the offers that they have accepted or rejected before. In Chae and Yang (1994), Krishna and Serrano (1996) and Suh and Wen (2006) the bargaining procedure encompasses bilateral negotiations. Yang (1992) modifies the game to make the offers include only the share of the last player in the responding order. In Merlo and Wilson (1995), the identity of the proposer is determined by a stochastic process. Chatterjee and Samuelson (1990) and Stahl (1990) consider simultaneous offers in the bargaining game, whereas Perry and Reny (1993) and Sakovics (1993) assume there might be lags before players recognize and respond to offers. In Baron and Ferejohn (1989), agreement does not require unanimous consent. Haller and Holden (1990) imposes costs on rejecting an offer.

Binmore, Shaked, and Sutton (1989) and Huang (2002) allow for outside options.

Asheim (1992) employs the notion of “acceptable paths”, where a path is acceptable

if and only if a player cannot profit by rejecting an offer an proposing another one

herself. Baliga and Serrano (2001) and Vannetelbosch (1999) allow players to exit

with partial agreements and the latter analyzes the equilibria using the refinement

of trembling-hand rationalizability.

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One method aiming to eliminate the multiplicity of equilibria in the multiplayer alternating offers bargaining game concerns the use of bounded rationality. This method is based on the idea that players prefer to use less “complex” strategies since learning and implementing them is easier. Different definitions of “complex- ity” have been used in this regard. The most basic, but just as debated, way of using

“non–complex” strategies is imposing stationarity. When players’ strategies are in- dependent of the past, Herrero (1985) shows that the subgame perfect equilibrium is unique, and corresponds to the cooperative solution in the multiplayer cooperative game. However, using stationary strategies in order to formulate bounded rationality is disputed due to its extremity. We will elaborate more on this subject in Chapter 4.

Another, less extreme, formulation is based on the number of states of an automa- ton representing a strategy. We refer the reader to Aumann (1981) for a detailed discussion about the use of strategies described by finite–state automata in the anal- ysis of bounded rationality. This particular “counting states” measure, on the other hand, does not provide a remedy. This is because, the strategy Shaked uses to prove that every feasible allocation can be obtained in subgame perfect equilibrium is a finite–state automaton (see Theorem 3 of the current thesis). It only uses n + 1 number of states, where n denotes the number of players. That is, this method does not solve the multiplicity of equilibria. Hence, Chatterjee and Sabourian (2000) uses a different formulation of complexity, which they define as: “...if two strategies are otherwise identical except that in some instances the first strategy uses more infor- mation than that available in the current “stage” of bargaining and the second uses only the information available in the current “stage”, then the first strategy is more complex than the second.”(Chatterjee and Sabourian (2000, p.1493)). However, al- though shrinking the set of Nash equilibria to some extent, they do not obtain a unique equilibrium either

1

.

In this study, we use an alternative and widely accepted notion of bounded ra-

1

Because that complexity is associated with some costs, they are obliged to employ modified

notions of subgame perfection as in Rubinstein (1986) and Abreu and Rubinstein (1988).

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tionality in the n–player discounted alternating offers bargaining game. Specifically, each player is restricted to use “one–memory” strategies. That is, players can base their actions only on yesterday’s history, and cannot recall the history prior to that.

Employing the restriction to one–memory strategies, we establish that the subgame perfect equilibrium outcome is unique and equivalent to the multiplayer general- ization of Rubinstein’s. This also implies that the unique subgame perfect equilib- rium outcome corresponds to the Nash solution in the multiplayer cooperative game.

Therefore, the main finding of this thesis constitutes a significant contribution to the Nash program.

It is important to note that, the notion of one–memory does not imply or is not implied by any of the other complexity formulations mentioned above. In fact, we show that the strategy used in Shaked’s Theorem (presented as Theorem 3 in the current thesis which establishes the multiplicity of subgame perfect equilibria) is not one–memory, although being associated with a finite–state automaton. More- over, Chatterjee and Sabourian (2000) is about complexity costs and not restricting history–dependence.

An interesting observation is that in the proof of our main result, we establish that one–memory subgame perfect equilibrium strategies have to be stationary. Indeed, the rest of our proof follows Herrero (1985, Proposition 4.2, p.90), which is restated (and proven for purposes of completeness) as Theorem 4 in the current thesis.

In the next chapter, we introduce our model and make the necessary definitions.

In Chapter 3, we provide an overview of the axiomatic approach to bargaining and

elaborate on some important theorems in non–cooperative bargaining that are rele-

vant to our main result, specifically those of Rubinstein’s and Shaked’s. In Chapter

4, we present our main result, in addition to some auxiliary results.

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2 The Model

The set of players is given by N = {1, . . . , n} and they are bargaining on the partition of a pie of size one. Only if they unanimously agree upon a partition will they get their respective shares. Otherwise, each will get none. The bargaining process is as follows: The game starts in period 0 with an offer

x = x

(i)



i∈N

∈ X by player 1, where

X ≡ y ∈ [0, 1]

n

: y

(1)

+ . . . + y

(n)

≤ 1 ⊂ R

n+

is the set of all offers. Then all other players respond one by one in the order 2, 3, . . . , n, by either accepting (Y ) or rejecting (N ) the offer. If all of them choose Y , the game ends and everybody gets her agreed share. If any of the players choose N , the game goes on to the next period, where the same process starts again with an offer by player 2 and responds by other players in the order 3, 4, . . . , n, 1. This goes on until there is agreement on an offer.

The set of all histories is denoted by H. The empty history is included in the

set of all histories, i.e. ∅ ∈ H. A one–period history is defined as a history end-

ing in the same period at the beginning of which it started. The set of all one-

period histories is denoted by H

1

≡ {{(x), (x, R)} : R = (r

1

, . . . , r

m

) where r

k

{Y, N } for k = 1, . . . , m and m ≤ n − 1}. Similarly, we define H

11

≡ {{(x), (x, R)} :

R = (r

1

, . . . , r

m

) where r

k

∈ {Y, N } for k = 1, . . . , m and m ≤ n − 2}, and H

21

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{{(x, R)} : R = (r

1

, . . . , r

n−1

) where r

k

∈ {Y, N } for k = 1, . . . , n − 1 and r

k0

= N for at least

one k

0

= 1, . . . , n − 1}, and finally H

31

≡ {{(x, R)} : R = (r

1

, . . . , r

n−1

) where r

k

= Y for all k = 1, . . . , n − 1}. In words, H

11

is the set of all one-period histo- ries after which one or more players are yet to play for that period to end. H

21

is the set of all one-period histories after which the period ends, and the next one starts. H

31

is the set of all terminal 1–period histories. A t–period history is denoted by h

t

= (h

0

, h

1

, ..., h

t−1

), where h

s

denotes the one-period history covering period s with h

s

∈ H

21

for all s ∈ {0, 1, . . . , t − 2} and h

t−1

∈ H

1

. H

t

is the set of all t-period histories. Following the above, we let H

1t

≡ {h

t

∈ H

t

: h

t−1

∈ H

11

}, and H

2t

≡ {h

t

∈ H

t

: h

t−1

∈ H

21

}, and finally H

3t

≡ {h

t

∈ H

t

: h

t−1

∈ H

31

}. The set of all terminal histories is denoted by Z = S

t=1

H

3t

.

The concatenation of two histories h

t

= (h

0

, . . . , h

t−1

) ∈ H

2t

and h

s

= (h

00

, . . . , h

0s−1

) ∈ H

s

is the operation defined as h

t

◦ h

s

= (h

0

, . . . , h

t−1

, h

00

, . . . ,

h

0s−1

) ∈ H

t+s

.

The 1–tail of a history h

t

is the history starting from and including the last complete one–period history in h

t

. Formally, the 1–tail of h

t

= (h

0

, h

1

, ..., h

t−1

) is defined as

T

1

(h

t

) =

h

t−2

◦ h

t−1

if h

t

∈ H

1t

h

t−1

if h

t

∈ H

2t

∪ H

3t

.

In this setting, a strategy for player i is a function f

i

mapping any non–terminal

history in which she decides (denoted by H(i)) into the actions allowed for her at

that particular history. The set of all strategies of player i is denoted by F

i

. As

usual, F = ×

i∈N

F

i

and F

−i

= ×

j6=i

F

j

. It should be emphasized that f

j(i)

(h

t

) denotes

the share of player i in the partition offered by player j in history h

t

, i, j ∈ N and

h

t

∈ H

2t

and t ∈ N where (t mod n) + 1 = i. A strategy f

i

is one–memory if for

all (h

t

, h

s

) with t, s ∈ N where (t mod n) = (s mod n) satisfying T

1

(h

t

) = T

1

(h

s

)

we have f

i

(h

t

) = f

i

(h

s

). The set of one–memory strategy profiles is denoted by

F

1

= ×

i∈N

F

i1

. Hence, a one–memory strategy assigns the same actions to two

histories that are different but have the same 1–tail. Notice that the two histories

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need not be of same length, i.e. the one–memory restriction does not allow time–

dependence.

Given a strategy f

i

∈ F

i

and a history h ∈ H we denote the strategy induced by f

i

at h by f

i

|h. Thus, (f

i

|h)(¯ h) = f

i

(h ◦ ¯ h) for every (h ◦ ¯ h) ∈ H(i). Let f |h = (f

i

|h)

i∈N

and F |h be the set of all such strategy profiles. A strategy profile f induces two types of terminal histories, the first where players agree upon some partition in a finite time period, and the second where this does not hold. Players use a common discount factor δ ∈ (0, 1). In the first type of terminal histories that involve an allocation x = x

(i)



i∈N

∈ X agreed upon in period t, the utility of player i, i ∈ N , is a function U

i

: F → R where U

i

(f ) is given by u

i

: X × N → [0, 1] which is defined by u

i

(x, t) = δ

t

x

(i)

. In case of no agreement –the second type of terminal histories– player i gets a utility U

i

(f ) given by lim

t→∞

u

i

(x, t) = lim

t→∞

δ

t

x

(i)

= 0.

A strategy vector f ∈ F is a Nash equilibrium if U

i

(f ) ≥ U

i

( ˆ f

i

, f

−i

) for all i ∈ N and

f ˆ

i

∈ F

i

. Also, f ∈ F is a subgame perfect equilibrium (SPE) if f |h ∈ F |h is a Nash

equilibrium for all h ∈ H. The set of all subgame perfect equilibrium strategy profiles

is denoted by E and those with one-memory by E

1

, i.e. E

1

= E ∩ F

1

. It should

be pointed out that, in the current study, the definition of one–memory subgame

perfect equilibrium does not involve only deviations that are one–memory. In fact,

there are no limitations on the magnitude of memory of a strategy that constitutes

a deviation from a one–memory strategy.

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3 The Bargaining Problem

In this chapter, first we will discuss Nash’s cooperative game theoretic solution to convex bargaining problems employing an axiomatic approach. Then, the well–

known result due to Rubinstein (1982), establishing the uniqueness of subgame per- fect equilibrium in two player alternating offers bargaining games (with discounting) will be presented. Moreover, it should be emphasized that the subgame perfect equi- librium outcome of two player alternating offers bargaining games (with discounting) converges to the unique Nash solution of the underlying bargaining problem with suf- ficiently patient players (i.e. when the common discount factor tends to 1).

In general, following Serrano (2005), the Nash program “is an attempt to bridge the gap between the two counterparts of game theory (cooperative and non-cooperative)”.

In fact, obtaining the Nash bargaining solution in a plausible non-cooperative game theoretic formulation as the unique (non-cooperative game theoretic) equilibrium constitutes a noteworthy contribution to the Nash program. Therefore, Rubinstein’s result is, indeed, a corner stone in the Nash program.

However, when Rubinstein’s setting is generalized to more than two players,

the above discussed and well celebrated result no longer holds. In fact, Shaked

(1986) shows that in the multiplayer bargaining game, also known as the Shaked’s

game, every feasible allocation can be sustained in subgame perfect equilibrium. To

make things worse, this striking result weakening game theoretic prediction power,

is obtained using finite automata strategies, which are the standard modeling tool

to capture bounded rationality. That is, the result in the Nash program with more

than two players is not established even when attention is restricted to the use of

non–complex (finite automata) strategies.

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3.1 Nash’s Barganing Solution

According to the framework established by Nash (1950), a bargaining situation con- sists of the set of players N = {1, 2}, the set of possible agreements A, the disagree- ment outcome D, and a preference relation 

i

for each player i ∈ N on lotteries over the set of possible agreements and the disagreement outcome, i.e. over A ∪ D.

Let the set of lotteries over A∪D be denoted by L = {` ∈ [0, 1]

|A∪D|

: P

|A∪D|

i=1

`

i

= 1}. Players’ preference relations satisfy von Neumann and Morgenstern assumptions.

Thus, the preference relation for each player i can be represented by a utility function v

i

: L → R, where a lottery is preferred by player i to another if and only if the expected utility of that lottery is greater than the expected utility of the other lottery according to v

i

.

Let S = {s ∈ R

2

: ∃` ∈ L with (v

1

(`), v

2

(`)) = (s

1

, s

2

)} denote the set of all possible utility pairs and let d = (v

1

(D), v

2

(D)). Nash defines a bargaining problem as the pair hS, di, where S ⊂ R

2

is compact and convex, d ∈ S, and there exists s ∈ S with s

i

> d

i

for i = 1, 2. Letting B denote the set of all bargaining problems, a bargaining solution is defined as a function f : B → R

2

, assigning a unique element of S to each bargaining problem hS, di.

1

The bargaining problem hS, di does not specify any details as to the nature of the bargaining process. Instead of dealing with the details of a given bargaining process, Nash employs an axiomatic approach. That is, he specifies four properties that he expects a natural solution to have, and defines the subset of S satisfying all four of these properties as the solution of the bargaining problem hS, di. We state these four axioms below (Osborne and Rubinstein 1990):

1. Invariance to equivalent utility representations: Suppose that hS

0

, d

0

i is ob- tained from hS, di using the transformation s

0i

7→ α

i

s

i

+ β

i

for some α

i

, β

i

∈ R, α

i

> 0, for i = 1, 2, i.e. S

0

= {(α

1

s

1

+ β

1

, α

2

s

2

+ β

2

) : (s

1

, s

2

) ∈ S}, and d

0

= (α

1

d

1

+ β

1

, α

2

d

2

+ β

2

). Then f

i

(S

0

, d

0

) = α

i

f

i

(S, d) + β

i

, for i = 1, 2.

1

Note that, f here is not the same as the f we used to denote strategies in our model, in

Chapter 2. Here, for the sake of convention, f denotes the bargaining solution. However, outside

this section, f will continue to stand for strategies.

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2. Symmetry: If the bargaining problem is symmetric, i.e. d

1

= d

2

and (s

1

, s

2

) ∈ S if and only if (s

2

, s

1

) ∈ S, then f

1

(S, d) = f

2

(S, d).

3. Independence of irrelevant alternatives: If hS, di and hT, di are bargaining prob- lems with S ⊂ T and f (T, d) ∈ S, then f (S, d) = f (T, d).

4. Pareto efficiency: Suppose hS, di is a bargaining problem and s, t ∈ S are such that t

i

> s

i

for i = 1, 2. Then f (S, d) 6= s.

In his seminal paper (Nash 1950), Nash establishes that the bargaining solution satisfying all four of these axioms is unique. We next present, without proof, this very well–known result:

Theorem 1 (Nash (1950)) There is a unique bargaining solution f

N

: B → R

2

satisfying the axioms 1-4, which is given by

f

N

(S, d) = arg max

(d1,d2)≤(s1,s2)∈S

(s

1

− d

1

)(s

2

− d

2

). (3.1)

It should be emphasized that when the Axioms 1 – 4 are generalized to the case of n–players, the bargaining solution satisfying all four axioms remains to be unique.

In fact, it is the n–player generalization of 3.1:

arg max

d≤s∈S n

Y

i=1

(s

i

− d

i

). (3.2)

Moreover, it is important to note that, 3.2 gives the solution as equal division due to Axiom 2 when the bargaining problem is symmetric, which is the case with the underlying bargaining problem in our model.

3.2 Rubinstein’s Result

We now turn our attention back to non-cooperative bargaining games, which are

the focus of this thesis, as described in our model. In this section, we present the

well-known result of Rubinstein (1982) and its proof, establishing the uniqueness of

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subgame perfect equilibrium in the 2–player discounted alternating offers bargaining game:

Theorem 2 (Rubinstein (1982)) The bargaining game with two players has a unique subgame perfect equilibrium f

, where (regardless of the history) player 1 proposes a sharing scheme given by x

=

1+δ1

,

1+δδ

 whenever it is her turn to pro- pose, and player 2 proposes y

=

1+δδ

,

1+δ1

. Each player accepts an offer if and only if it delivers a share not less than

1+δδ

.

Proof. First, we need to show that the strategy profile denoted by f

and specified in the statement of the Theorem, constitutes a subgame perfect equilibrium.

Consider a subgame starting with an offer by player 1. If player 1 proposes x 6= x

, then either x

(2)

>

1+δδ

, and x gets accepted, where player 1 will receive x

(1)

<

1+δ1

, or x

(2)

<

1+δδ

, which will result in the rejection of x and acceptance of y

the next period, providing player 1 with

1+δδ

with one period of delay. Thus, the highest utility player 1 can get, by deviating from x

is less than x

∗(1)

. Now suppose player 2 deviates from choosing Y in the subgame starting after the offer x

by player 1. After rejecting x

, player 2 will propose y

, and it will be accepted. Player 2 will get

1+δ1

with one period of delay, which gives her a utility equal to getting x

∗(2)

with no delay. Hence, no player has a profitable deviation. Clearly, the same arguments apply to the subgames starting with an offer by player 2, or a response by player 1, establishing that f

is a subgame perfect equilibrium.

Now, we need to show that f

is the unique subgame perfect equilibrium. Let G

i

denote a subgame starting with an offer by player i. Since it is shown above that the set of subgame perfect equilibria is not empty, we can define:

M

i

≡ sup{δ

t

x

(i)

: there is an SPE of G

i

with outcome (x, t)}

m

i

≡ inf{δ

t

x

(i)

: there is an SPE of G

i

with outcome (x, t)}.

In the following steps, we will establish that the present utility that player 1 gets

from every SPE outcome of G

1

is x

∗(1)

, and the present utility that player 2 gets

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from every SPE outcome of G

2

is y

∗(2)

, i.e.

M

1

= m

1

= x

∗(1)

and M

2

= m

2

= y

∗(2)

. (3.3)

Step 1 m

2

≥ 1 − δM

1

.

Proof. Suppose player 2 proposes y with y

(1)

> δM

1

. Player 1 will get at most M

1

with one period of delay if she rejects, so she will accept any such offer y. Thus, the minimum utility that player 2 gets in a SPE of G

2

must be 1 − δM

1

.

Step 2 M

1

≤ 1 − δm

2

.

Proof. By rejecting an offer by player 1 in G

1

, Player 2 can guarantee getting at least m

2

the next period. Thus, in any subgame perfect equilibrium of G

1

with no delay, player 2 must get at least δm

2

, which means player 1 can get at most 1 − δm

2

. If player 2 rejects the offer by player 1 and agreement is delayed, player 1 gets at most 1 − m

2

with one period of delay. Since δ(1 − m

2

) ≤ 1 − δm

2

, the result follows.

Multiplying both sides of Step 1 by δ, and then subtracting both sides from 1 gives 1 − δm

2

≤ 1 − δ + δ

2

M

1

. This, together with Step 2 implies M

1

≤ 1 − δ + δ

2

M

1

, which is equivalent to M

1

1+δ1

. We know that M

1

≥ x

∗(1)

=

1+δ1

since f

is SPE.

Hence, it must be M

1

= x

∗(1)

. Similarly, multiplying both sides of Step 2 by δ and subtracting from 1 gives 1 − δM

1

≥ 1 − δ + δ

2

m

2

, which, together with Step 1 implies m

2

1+δ1

. Since it must be that m

2

≤ y

∗(2)

=

1+δ1

, we get m

2

= y

∗(2)

.

Replacing player 1 with player 2 in the steps and the following argument above will give m

1

= x

∗(1)

and M

2

= y

∗(2)

.

The remaining part is to show that in any subgame perfect equilibrium, the first

offer is accepted. Suppose, to the contrary, player 2 rejects player 1’s first offer. The

next period, an SPE of G

2

will be followed. By 3.3, the present value (in G

2

) of such

an outcome to player 2 is y

∗(2)

. Thus, the present value (in G

2

) of the same outcome

to player 1 is no more than y

∗(1)

, which makes the present value to player 1 in G

1

,

δy

∗(1)

< x

∗(1)

resulting in a contradiction with 3.3. Hence, in any SPE, the first offer

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must be accepted, finishing the proof.

Formally, the strategy profile f

employed in the proof of Rubinstein’s result is defined as the following:

Let z

i

, i ∈ N be such that z

i(i)

=

1+δ1

and z

i(−i)

=

1+δδ

, i.e. z

1

= x

and z

2

= y

; x

, y

as specified in the statement of Theorem 2. Then for i ∈ N and for any given history h

t

= (h

0

, . . . , h

t−1

),

f

i

(h

t

) =

 

 

 

 

 

 

z

i

if i = t mod 2 + 1, Y if i = t mod 2,

and h

t−1

= z with z

(i)

1+δδ

N otherwise.

(3.4)

The remark below emphasizes an important observation about f

. Remark 1 f

defined by 3.4 is stationary.

3.3 The Multiplayer Case

The result of Rubinstein (1982) (Theorem 2) does not hold when the number of players is more than 2. In fact, Shaked (1986) shows that in the 3–player bargaining game, also known as the Shaked’s game, every feasible allocation can be sustained in subgame perfect equilibrium. This is presented in the following theorem by Shaked, restated as Theorem 3.4 in Osborne and Rubinstein (1990):

Theorem 3 (Shaked (1986)) When δ ≥

12

, for any partition x

∈ X, there exists a subgame perfect equilibrium of the 3–player bargaining game, where x is agreed upon in period 0.

Proof. Following the same notation with Osborne and Rubinstein (1990), we

consider the strategy profile σ

represented by automata in Table 3.3. In words,

the strategy profile σ

prescribes each player to propose y at state y and accept an

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x

e

1

e

2

e

3

1 proposes x

e

1

e

2

e

3

accepts x

1

≥ δx

1

x

1

≥ δ x

1

≥ 0 x

1

≥ 0

2 proposes x

e

1

e

2

e

3

accepts x

2

≥ δx

2

x

2

≥ 0 x

2

≥ δ x

2

≥ 0

3 proposes x

e

1

e

2

e

3

accepts x

3

≥ δx

3

x

3

≥ 0 x

3

≥ 0 x

3

≥ δ

Transitions

If in any state y, any Player i proposes x with x

i

> y

i

, then go to state e

j

, where j 6= i is the player with the lowest index for whom x

j

< 1/2.

Table 3.1: Shaked’s SPE strategy profile σ

.

offer x if and only if x

i

≥ δy

i

. The game starts at (equilibrium) state x

, and the state changes whenever (and immediately after) player i proposes z with z

i

> x

i

, i.e. whenever she proposes a share to herself exceeding the one she is supposed to obtain in that particular state. In that case, transition occurs to the new state e

j

where j is the player with the smallest index for whom player i has proposed a share less than 1/2. Indeed, e

j

constitutes the reward state of player j. Notice that, player j will reject z, since the transition to state e

j

occurs immediately after z is offered, and the strategy of player j prescribes her to reject z with z

j

< δe

jj

= δ at the new state e

j

. The state does not change if a player deviates from accepting or rejecting an offer.

These strategies constitute a subgame perfect equilibrium. To see this, take any state y, y ∈ {x

, e

1

, e

2

, e

3

} and player i, i ∈ {1, 2, 3}. At this state, if player i proposes x with x

i

> y

i

, the state changes to some e

j

with j 6= i, j ∈ {1, 2, 3}, which means that x will be rejected and the next period e

j

will be offered and accepted, giving player i a payoff of 0. If player i proposes x with x

i

< y

i

, either it will get accepted and she will get x

i

, or it will get rejected and y will be accepted the next period, giving player i, y

i

with one period of delay. In both cases, player i gets a present utility less than y

i

, so it is not optimal for player i to deviate.

Now suppose at state y, it is player i’s turn to respond to the offer. If she rejects

y, the next period y will be offered again, and accepted. Player i will get y with one

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period of delay, making the deviation unprofitable. Hence, σ

is a subgame perfect equilibrium.

3.4 Finite–State Automata and Bounded Ratio- nality

It is important to note that the strategy profile σ

described in Table 3.3 consists of a finite–state automaton for each player.

Finite–state automata have been used extensively in the literature as a means to model complexity of strategies and bounded rationality of players, an idea first proposed in the economics literature by Aumann (1981) following Simon’s pioneering formulations. Using such a formulation, Neyman (1985) shows that cooperation can be sustained even in finitely repeated games when the strategies are restricted to finite–state automata. Rubinstein (1986) and Abreu and Rubinstein (1988) impose costs on maintaining an additional machine state, rather than taking finite–state automata as given. The reasoning behind using finite–state automata is that players face a tradeoff between maximizing their payoffs in the game and using as simple strategies as possible. The less number of states an automata has, the less likely it is to break down, the easier it is to learn, and it requires less time and effort to be implemented (Rubinstein 1998).

Hence, Shaked’s Theorem (Theorem 3, employing a strategy profile described by the finite–state autamata given in Table 3.3) establishing that all feasible allo- cations can be sustained in subgame perfect equilibrium even when the strategies are restricted to finite–state automata bears a special significance. That is, the set of subgame perfect equilibria does not shrink even when players are assumed to be boundedly rational, bounded rationality being modeled as usual in the literature.

Counting the number of states of an automaton associated with a strategy is not

the only method to measure bounded rationality. In fact, it should be emphasized

that finite–state automata are less complex to implement compared to their infinite–

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such a strategy is simpler. However, in order to identify the state of the game, players may need to know the entire history. Therefore, finite–state automata do not restrict, in any way, the history–dependence of strategies, and may require players to “recall ” the entire history of the game. Clearly, this results in an inconsistency with the essence of complexity and bounded rationality. Hence, another method of alleviating such complexity involves finite–memory strategies.

3.5 The Loss of Game Theoretic Prediction Power

Rubinstein’s result (Theorem 2) that in 2–player alternating offers bargaining games with discounting, the subgame perfect equilibrium outcome is unique and converges to the equal division as the discount factor tends to 1 bears significance due to a number of facts. One of these is that the uniqueness of the subgame perfect equilibrium outcome raises the prediction power of the model. Another is that the unique outcome is plausible, and in fact, desirable. This is due to not only fairness concerns, but also the fact that, the game precisely implements the Nash bargaining solution in subgame perfection and hence provides it with a non–cooperative game theoretic foundation.

However, these desirable properties are not carried over to the game with n ≥ 3 players. Shaked’s result that every feasible allocation can be sustained in subgame perfect equilibrium means that we have no game theoretic prediction power as to what the equilibrium outcome of the game will be. Moreover, the Nash bargaining solution for n ≥ 3 players, which is equal division of the pie, is not implemented in subgame perfection in an alternating offers bargaining game with n ≥ 3 players. In turn, this means that the Nash program fails to deliver a desirable result with n ≥ 3 players.

The emerging inconsistency between the conclusions of the 2–player and multi-

player versions of this game, has been well–known in the literature. For more on this

subject, we refer the reader to Osborne and Rubinstein (1990). In fact, many studies

consider modifications to the multiplayer version of the game at hand in order to

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obtain a unique equilibrium that can be associated with the Nash bargaining solu- tion. In that regard, some of the related studies are (but not restricted to) Binmore, Rubinstein, and Wolinsky (1986), Jun (1987), Chae and Yang (1988), Baron and Ferejohn (1989), Binmore, Shaked, and Sutton (1989), Chae and Yang (1990), Stahl (1990), Chatterjee and Samuelson (1990), Asheim (1992), Yang (1992), Perry and Reny (1993), Chae and Yang (1994), , Krishna and Serrano (1996), Vannetelbosch (1999), Chatterjee and Sabourian (2000), Baliga and Serrano (2001), Huang (2002), Lee and Sabourian (2005), and Suh and Wen (2006). While some of these modifica- tions feature plausible economic insight, only a few of them are concerned with the elimination of the inconsistency at hand by employing bounded rationality concerns.

When considering complexity formulations that are aimed to re-evaluate / allevi- ate these inconsistencies, the standard complexity formulations (based on counting the number of states of an automaton describing a player’s strategy), do not have any bite, as the above discussion displays. Therefore, the search for the elimina- tion of the inconsistencies between the 2–player and multiplayer versions based on bounded rationality concerns, necessitates considerations of nonstandard complexity formulations. Indeed, one such study is Chatterjee and Sabourian (2000).

It is useful to remind the reader that the current study considers a complexity

formulation based on restricting players’ memory (recall), the number of consecutive

periods that a player can recall. While it is “a” particular measure of complexity, it

needs to be emphasized that (to our knowledge) this is the pioneering work regarding

the use of memory in alternating offers bargaining games.

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4 One–memory

The sharp contrast between the results of the 2–player and multiplayer versions, i.e. the inconsistency presented in the previous Chapter, ruining game theoretic prediction power in the multiplayer version, is due to the fact that in the multiplayer case one of the players can be rewarded for rejecting a deviant offer. This is an observation that does not hold in the 2–player case. In fact, this interesting, yet problematic, contrast holds not only when players’ strategies are unconstrained (i.e.

players are fully rational) but also when players are boundedly rational (i.e. players’

strategies are constrained to finite–state automata).

As mentioned in Section 3.4, the problem with using finite–state automata or complexity costs such as in Chatterjee and Sabourian (2000) to model bounded rationality is that they may require players to recall the entire history of the game:

Even though each player’s strategy (described by a finite–state automaton) is, by definition, a function from a finite set of states into that player’s set of actions, the identification of the state that the game is in, may necessitate to employ the entire history of the game, requiring players to have unbounded memory (recall) in turn.

This, clearly, is unappealing. For more on this subject, we refer the reader to Barlo, Carmona, and Sabourian (2012).

One remedy proposed to solve this issue consists of the use of stationary strategies.

However, that particular method is arguably not that appealing. This is because,

the difference between using strategies described by finite–state automata and sta-

tionary plans of actions would amount to going from the situation where a player

is concerned about the number of states associated with the automaton describing

her strategy, into one where she has to use a strategy that does not depend on any

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past information. Moreover, as Osborne and Rubinstein (1994) argue, it is difficult to justify the use of stationary strategies only based on their simplicity, considering that the equilibrium should also be an equilibrium in beliefs and there is no reason for a player to believe that others will choose the same action, even when many deviations have occurred before.

Thus, limited memory strategies present themselves as a natural way to model bounded rationality without being too restrictive; and, when extending the notion of stationarity to allow history dependence, the next immediate and evident formulation in the current setting involves the use of one–memory

1

.

In this chapter, we consider a widely accepted method of restricting players’

strategies on grounds of bounded rationality. Particularly, all players are assumed to use one–memory strategies, plans of actions that depend only on yesterday’s profile of choices. It is appropriate to point out that in this setting, each player can condition her actions not only on the current offer, but also on the outcome of the last stage (period) of the game. This stage includes both the offer proposed in that particular period and associated responses made by others. It is also imperative to mention that deviations from one–memory strategy profiles are not restricted to be of one–

memory. That is, (unlike Rubinstein (1986) and Abreu and Rubinstein (1988)) we do not use a modified notion of subgame perfection, but employ the standard one.

For more about this distinction, we refer the reader to Kalai and Stanford (1988).

Furthermore, when employing one–memory strategies, a player cannot condition her actions on calendar time. At this stage, it is appropriate to point out that the notion of memory employed in this study is the one–period version of those that are used in the following studies: Sabourian (1989), Sabourian (1998), Barlo and Carmona (2007), Barlo, Carmona, and Sabourian (2009), and Barlo, Carmona, and Sabourian (2012). And it differs from the time–dependent versions of Barlo (2003), Cole and

1

Notice that the notion of Markov perfect equilibria, subgame perfect equilibria where players’

strategies are restricted to be functions obeying the Markov property, are generally considered the

next step in extending stationarity to allow for history dependence. However, a Markov strategy

necessitates the use of a finite state space, hence, is nothing but a finite–state automaton. There-

fore, as the Chapter3 has depicted, such an extension does not eliminate the inconsistency under

consideration.

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Kocherlakota (2005), H¨ orner and Olszewski (2006), H¨ orner and Olszewski (2009), Mailath and Olszewski (2011).

In what follows, first we show that the strategy profile σ

(see Table 3.3) used in the proof of Theorem 3 is not one–memory. It is worthwhile to remind the reader that Theorem 3 establishes that for any feasible partition of the pie, there exists a subgame perfect equilibrium strategy profile where each component strategy is described by a finite–state automaton. That is, σ

is described by finite–state automata, yet is not one–memory.

Following this, an auxiliary result due to Herrero (1985) is presented and proven to maintain completeness of this thesis: The stationary subgame perfect equilibrium is unique and in the multiplayer version, corresponds to the multiplayer generaliza- tion of Rubinstein’s.

We prove that the one–memory subgame perfect equilibrium outcome displays the very same properties as the stationary one: It is unique, and in the multiplayer version, it corresponds to the associated generalization of Rubinstein’s result.

Our main result constitutes a noteworthy contribution to the Nash program be- cause it provides a remedy based on a widely accepted notion of bounded rationality, one–memory, eliminating the inconsistency between the conclusions of the 2–player and multiplayer versions.

In the following section, we elaborate on the main ingredients of our result, includ- ing a particular notion referred to as confusion by Barlo, Carmona, and Sabourian (2009). Moreover, Sections 4.2 and 4.3 provide some needed auxiliary results, while Section 4.4 presents our main contribution.

4.1 Confusion

When attention is restricted to the standard discounted repeated games, one–memory,

the initial step in extending the notion of stationarity to allow history dependence,

delivers striking results. In fact, an immediate observation in this setting is that the

set of stationary subgame perfect equilibria consists of strategy profiles that involve

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the repetition of a given Nash equilibrium of the stage game. That is why stationary subgame perfect equilibria are not interesting in this setting. On the other hand, Barlo, Carmona, and Sabourian (2009) establishes that the subgame perfect Folk Theorem holds with one–memory strategies, when the set of actions in the stage game of the repeated game is sufficiently “large” for each player, so that each payoff profile is not isolated. Hence, in such games the set of subgame perfect equilibria gets considerably richer when players are allowed to use even one–memory. The next ob- servations about the one–memory Folk Theorem of Barlo, Carmona, and Sabourian (2009) should be pointed out:

The large action space assumption is critical in establishing these results because it allows players to encode the entire history of the past into the previous period’s actions. More formally, with rich action sets any equilibrium strategy vector in which each player strictly prefers not to deviate at every history, can be perturbed so that each player chooses different actions at different histories. With such distinct plays of the game, at each date the players can use the outcome of the previous period to coordinate their actions appropriately. Thus, the original equilibrium can be approximated by another that has one period recall. (Barlo, Carmona, and Sabourian (2012, p.1))

While one–memory strategies are a considerably rich set of plans of actions, they also bring about an interesting problem identified in Barlo and Carmona (2007) and elaborated on Barlo, Carmona, and Sabourian (2009) and Barlo, Carmona, and Sabourian (2012): Confusion. With one–memory, problems of detecting the latest deviation and the identity of the deviator arise.

Indeed, the notion of subgame perfection is equivalent to the principle of one–

deviation

2

(see Rubinstein (1982), Abreu (1988), and Osborne and Rubinstein (1994)) under certain assumptions which are satisfied in the setting considered in this thesis.

In fact, the use of discounting in the current analysis provides the critical argument to that regard. It requires the following two critical properties: First, a single player deviation can be detected and second, the identity of the deviator can be revealed.

If either of the above two properties were not to hold, there may be incentives for

2

It is worthwhile to point out that the principle of one–deviation is also referred to as the

principle of optimality or the single deviation principle.

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some player to deviate and manipulate the path of future play. In Barlo, Carmona, and Sabourian (2009), this is referred to as “confusion-proofness”.

These issues, do not arise with unbounded memory because one can use induction starting from beginning of the game to identify the latest deviation. With bounded memory, this inductive reasoning is not feasible. This, indeed, constitutes the main reason why a strategy described by a finite–state automaton needs not be finite–

memory.

In order to see the issue of confusion in the scope of standard repeated games, the following example from Barlo and Carmona (2007) is appropriate:

Consider a discounted repeated Prisoners’ Dilemma in which at every date each player can either cooperate C or defect D, and suppose that we want to implement a cycle consisting of

((C, D), (D, C))

yielding an average payoff strictly higher than the repetition of (D, D) forever. The strategy inducing this cycle, denoted by π = {π

t

}

t=1

, and involving the play of (D, D) forever for any history inconsistent with the equilibrium path, is subgame perfect with unbounded memory and with sufficiently high discount factors. However, the limited–memory versions of this strategy is not subgame perfect regardless of the magnitude of memory. This is because, if players can remember at most M periods, one player prefers to deviate at a history with its last M entries equal to (a

1

, π

2

, . . . , π

M

) with a

1

6= π

1

instead of playing the punishment: If π

M

= (D, C), then player 1 can play C instead of D, which will make the play return to the equilibrium outcome in the next period. Notice that if π

M

= (C, D) by a similar argument, player 2 would deviate. Hence, player 1’s continuation payoff in that history strictly exceeds the payoff he would receive by not deviating. (Barlo and Carmona (2007, p.3))

It is useful to point out that the issue of confusion, while creating cumber- some obstacles, is not sufficiently strong in the analysis of standard discounted re- peated games to make the subgame perfect Folk Theorem fail: Barlo, Carmona, and Sabourian (2009) prove the subgame perfect Folk Theorem with one–memory strategies and with rich action spaces in the stage game.

An interesting observation about alternating offers bargaining games is that while

not being a discounted repeated game in the standard sense, the stage game still

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involves a rich action space. In fact, in the alternating offers bargaining game (re- gardless of the number of players), the proposer has the option to encode the entire history of the game into his offer at a negligible cost. In turn, this also establishes the observation that the set of one–memory strategies is considerably richer than the set of stationary strategies

3

.

In the analysis of the alternating offers bargaining game with discounted utilities, confusion resulting from the one–memory requirement eliminates the multiplicity of equilibria featured in Shaked’s Theorem, Theorem 3. It is imperative to indicate that this observation holds even though the entire history can be recorded into players’

offers.

In what follows, we elaborate on how confusion, brought about by the one–

memory requirement, eliminates all but the stationary subgame perfect equilibrium.

Indeed, what we prove is that one–memory subgame perfect strategies have to be stationary, which together with Herrero (1985) implies our main finding

4

.

The first step which is formally presented in Lemma 3, shows that with subgame perfection and one–memory, each player always proposes the same share to herself regardless of the past. In order to see this in a 3–player version (for the sake of simplicity), consider the beginning of the game (the unique history with time period 0) and any history in time period 3. In both of them player 1 proposes. If player 1 were to obtain a strictly higher payoff in the 3–period history, she could deviate and offer the very same allocation (prescribed by her strategy for that particular 3–period history) in the beginning of the game. This deviant act can be identified by the other players as a single player deviation. Yet, if player 2 were to obtain a higher payoff by rejecting that offer at the beginning of the game, she would have rejected that very same offer in that 3–period history as well, which follows from

3

To see this, consider a finite time period and a given nonterminal history in the alternating offers bargaining game for which all the past offers were given by rational numbers that end in finitely many digits. Then, the proposer at this stage has the option to come up with an offer into which the entire history of the game can be recorded after a sufficiently high digit, hence, at a negligible cost. Therefore, having a large action space may provide players with the capacity to record the whole history of the game by only using one–memory.

4

Recall that Herrero (1985) establishes that the stationary subgame perfect equilibrium is unique

and in the multiplayer version corresponds to the multiplayer generalization of Rubinstein’s.

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the one–memory requirement: The last stage of the history after rejecting player 1’s deviant offer in the beginning of the game, is the same as the last stage of the history after rejecting player 1’s equilibrium offer in that 3–period history. In other words, seeing player 1’s deviant offer in the beginning of the game, player 2 does not find it profitable to punish her because she knows that tomorrow she will be confused as to which history they have been in, and hence, cannot be rewarded for punishing player 1. The same reasoning presented for player 2 holds for player 3 as well. Therefore, player 1 has to offer the same share to herself in both of these histories in any subgame perfect equilibrium. Clearly, following a similar reasoning for other histories and players establishes the observation.

The second step, presented in Lemma 4, displays that each player must propose the same offer in every history when employing the notion of subgame perfection and the requirement of one–memory. In other words, the part of the strategy involv- ing offers is stationary, which easily implies that the part of the strategy involving responses is also stationary. Hence, establishing the second step finishes the proof with the help of Herrero (1985).

To see the intuition behind the second step in a 3–player version of our game,

consider two histories both in which player 1 proposes, and in one of which player

3 gets a strictly higher payoff than the other one. Since both of these offers must

provide player 3 with payoffs that exceed the level that she can obtain by waiting

for 2 periods and proposing herself (which is fixed, as established by the first step),

the offer giving player 3 a strictly higher payoff than the other, also must provide

her a strictly higher payoff than the present value of the fixed amount that player

3 would get by waiting 2 periods and proposing herself. Now, consider the history

where player 3 is supposed to get a strictly higher payoff than the other history and

that player 1 cut player 3’s slack off her share (in a sufficiently small manner) and

distributed it between herself and player 2. At this history, observe that player 2

gets a strictly higher share than what she would get by rejecting. This is because,

by the first step, the next period she will propose herself the same share no matter

what the history has been. Since her share in the original offer (before player 1

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cut the slack off player 3’s share) has to be higher than the present value of what she would propose to herself in the next period, her share in this new offer must be strictly higher than what she would obtain by waiting one period and proposing herself. Thus, player 2 will be better off in this history if the offer gets accepted. For this to happen, player 3 must also accept, which will happen whenever she gets a lower payoff in the history after she rejects. However, since player 2 will be proposing in the next period, the best response property of her strategy (implied by subgame perfection) requires that she offers player 3 a lower payoff whenever player 3 rejects player 1’s offer in which player 2 is strictly better off. This general observation is established in a separate lemma, Lemma 5, for n ≥ 3 players.

It is also useful to notice that, the role that one–memory plays here, is that player 2 does not find it profitable to punish player 1 for deviating, because she knows that the next period she will be confused as to the identity of the deviator.

In fact, as established in the first step, she will propose the same share for herself next period whatever the history of this period turns out to be. Player 3 also does not punish player 1, since the next period, player 2 will propose only based on the last period’s history, which means that player 3 must make her decision considering how her actions will change that one–period history upon which player 2 will base her proposal the next period.

Therefore, by Lemma 5 the best response property of player 2’s strategy requires

the payoff that player 3 gets by rejecting player 1’s offer to be less than player 3’s

share in player 1’s offer, delivering the conclusion that in the history where player

1 cut the slack off player 3’s share and distributed among herself and player 2,

the offer must be accepted. Then, however, at the history where player 3 gets a

strictly higher share, distributing player 3’s slack between herself and player 2 is a

profitable deviation for player 1, establishing that there cannot be a subgame perfect

equilibrium where player 3 gets two different shares at two different histories both

in which player 1 proposes. The same argument applies to player 2’s shares, and to

histories where player 2 or 3 propose. Hence, each player always proposes the same

offer regardless of the history.

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In the rest of this chapter, we present this and some other results in a formal manner.

4.2 Shaked’s Strategy is not One–Memory

We show that the subgame perfect equilibrium strategies in the multiplayer bargain- ing game, described in Theorem 3 are not one-memory. Again, as was the case in the statement of Theorem 3, the result and the proof are presented for the case of three players, in order to avoid nonfruitful technicalities. Moreover, it should be noted that generalizing these results to n ≥ 3 players is trivial.

Proposition 1 The subgame perfect equilibrium strategy profile σ

given in Table 3.3 is not one-memory.

Proof. Let x

6= e

1

and x

6= e

2

. Take two histories

h = ((x

, N, Y ), (e

2

, N, N )) and h

0

= ((e

1

, N, N ), (e

2

, N, N )).

After history h, the game is at state e

1

, so σ

3

(h) = e

1

, whereas after h

0

, the state is e

2

and σ

3

(h) = e

2

. Since T

1

(h) = T

1

(h

0

) but σ

3

(h) 6= σ

3

(h

0

), σ

is not a one–memory strategy profile when x

6= e

1

and x

6= e

2

.

We need to check for x

= e

1

and x

= e

2

separately. First, we will check for x

= e

1

. Let

h

1

= ((x

, Y, N ), (e

2

, N, N ), (e

3

, N, N )), and

h

2

= ((x

, Y, N ), ((1/2, 1/2, 0), N, N ), (e

3

, N, N )).

Although T

1

(h

1

) = T

1

(h

2

), we have σ

1

(h

1

) = e

1

6= e

3

= σ

1

(h

2

), which means σ

is not one–memory when x

= e

1

.

The last case to check is x

= e

2

. Take the histories

h

3

= ((x

, Y, N ), (x

, N, N )) and h

4

= (((1/2, 1/2, 0), N, Y ), (x

, N, N )).

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